Question

The direction cosines of a line equally inclined to all the three rectangular co-ordinate axis are

• A. $\sqrt{3},\sqrt{3},\sqrt{3}$
• B. $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
• C. $1,1,1$
• D. None of these

Answer 1

The correct option is B

$\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

Explanation for the correct option:

Find the direction cosines :

Given that direction cosines of a line equally inclined to all the three rectangular co-ordinate axis.

Let the direction cosines of the line make an angle $\alpha$ with each of the coordinate axes.

So, the direction cosines of each axis are given by
$l=\cos \alpha ,m=\cos \alpha ,n=\cos \alpha$ and

we know that ,the property of direction cosines as
$l^2+m^2+n^2=1$

By putting the required values and solving for $\alpha$ we get
$$\begin{gathered} {\cos }^2\alpha +{\cos }^2\alpha +{\cos }^2\alpha =1 \\ 3{\cos }^2\alpha =1 \\ {\cos }^2\alpha =\frac{1}{3} \\ \cos \alpha =\pm \frac{1}{\sqrt{3}} \end{gathered}$$
The direction cosines are $\pm \frac{1}{\sqrt{3}}+\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}$

Therefore, by the given options we conclude direction cosines to be $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
Hence, option (B) is the correct answer.